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Lesson 4 – Applying Our Understanding Of The Unit Of Measure – Discussion
Posted by Jon on December 6, 2019 at 5:22 amWhich example from this lesson highlighting why the unit of measure matters resonated with you the most?
Can you think of another example where you can apply unitizing and the idea of the unit of measure to break down a concept from a grade or course you teach?
Share your reflection below along with any wonders you still have.
Kyle Pearce replied 2 weeks ago 39 Members · 53 Replies 
53 Replies

All of the examples really highlighted the importance of how using units can help make connections for students (and teachers). I teach grade 5 and the struggle is always around teaching fractions. It’s difficult for a majority of the students to understand that fractions and decimals are related because they aren’t shown how they are connected. The modeling of division with fractions was really helpful. I can’t wait to show my PLC.

I love the visual of dividing fractions. In a recent math PD with grades 35, creating a visual model for fractional division was the biggest hang up for teachers.

Fractions are quite abstract for many students. I really want to use more concrete and visual representations of process. Unit of measure is especially important. I want to focus on getting students to articulate exactly what their answer means (reverse expression of etc.). The reciprocal is embedded into many concepts (graphing, proportional reasoning etc.) and is often a big downfall. Back to basics to build things up

Well. Even though I have taught various grade levels of Math (depending on the year – anything from K to gr. 10) I have never seen this type of visualization of division of fractions before. To be honest, I need to go over it again – i find it rather confusing!! (Hey, I was the memorization queen of algorithms in high school.) So i guess this is the lesson example that resonated with me the most. So much to learn!! Is there somewhere I can access more of this – trying to wrap my head around it.

Even though I’ve taught math (and fractions) for years, I am learning so much. I really appreciate the way you are breaking it down into small pieces of learning. Do you think it is important to use the word “partition” with students when you talk about breaking a whole into parts? I, too, confess to being confused by the visual for the partitive division of 3/4 divided by 1/4. I don’t think of it that way, and so it is hard to wrap my mind around the value of that particular method. Is there anything you can add? Maybe, as I progress through the lessons, it will become clearer to me. Thank you so much for this course!


Thanks for sharing!
What I always say to students who say something similar (ie: not sure if I get it or if I’m confused) is to try and convince me…
So… can you convince us? What does your model represent?


I will need to rewatch the division example. Now I see why teacher guides recommend using the language of “units” when working with base ten blocks. My primary colleges often call the smallest units “ones”. As a result, my grade five kids have difficulty conceptionally visualising the little red cubes as tenths of a rod. Playing with connecting cubes in groups of five would be a gradual first step. They love building things. (gun, towers, strange patterns) Your right. Adults are too quick to express abstractly with symbols. I will slow down.

Great points here.
Division is really interesting and ever since I came to the learning that there are two types of division, my mind has been opened to a completely new world! You may need many rewatches and you’ll likely need to do many problems yourself to really tinker with the two types and become fluent.


I agree that this is going to take some rewatching to fully understand the difference between partitive and quotative division. When I did a little searching to find some more examples of each type of division to help me better understand …. low and behold, look what I came across: https://mathisvisual.com/division/ !! A very helpful and visual explanation (perhaps using whole numbers helped my simple brain too! 😂) 8 divided by 2 as 8 divided into group of 2 or 8 divided into 2 groups.

Keep at it Lauren. It took some time and different examples for my high school teacher brain to get wrapped around the idea.


I love the idea of beginning with everything visually. How powerful it is for students to see that they’re dealing with units, which can be flexible, and it leads so beautifully from adding fractions to multiplying fractions, all with meaning and understanding.
Questions: how do I give my middle schoolers these experiences? how do I guide them through problems like this visually? how do I alleviate the internal hang ups from the years they’ve spent struggling with these concepts? how do i “undo” the tooquicktoabstract teaching they’ve been experiencing?

Great questions and I think it all comes back to thinking about what experience you’re hoping students will have and then asking “how can I offer this in a way that makes students curious?”
Often times this means using context, but some times it is just about a visual that makes you go “hmmm?!?!”
Have you checked out our Building Resilient Problem Solvers course in the academy and the curiosity path? That might serve as a good starting point.

Fractions as units of measure. I thinking deeply about how to engrain this into my middle schoolers so that they finally stop trying to add denominators and/or relying on tricks taught in previous grades like KFC (keep flip change) to deal with fractions without any idea about what they are doing.

I have found that it can take middle schoolers some time to adjust to the idea of using visual models after being exposed to mostly numbers and memorization in previous years. There can also be some confusion about what the bars mean. One bar with three orange boxes means three whole units and another bar with five boxes means each box equals one fifth. With my sixth grade, I’ve actually given them tiles and had them make the boxes in 3D before having them draw the process on paper. (so much harder to do when they’re online.)

The visual illustrations of both partitive and quotitive (measurement) division were very helpful. As a former curriculum coordinator one of the highlights of my experiences was working with parents who were often skeptical about the “new” ways math was being taught. To make a point for conceptual understanding vs. rote memorization of an algorithm, I would ask them if they recalled how to divide fractions. Often they would come up with “invert and multiply and don’t explain why.” Then we would have some fun with pattern blocks to demystify the why. Using a red trapezoid and comparing it to the yellow hexagon, they would identify it as a half of the yellow. Then we identified the blue rhombus as thirds. Now, let’s see how that algorithm works by taking the blue rhombii to determine how many can fit in the red trapezoid. Here it is demonstrated on BrainingCamphttps://www.youtube.com/watch?v=g5aET6HLMeI To illustrate a measurement division problem we would ask how many green triangles does it take to make onehalf using pattern blocks? https://youtu.be/oewp2S51FW4
In researching technology as a tool to teach division of fractions Conceptua Math is awesome. Click on link and you can see a demo using the line model http://www.conceptuamath.com/mathtools/

I enjoyed the example that illustrated multiplying fractions. Making use of the commutative property of multiplication, I challenged myself to illustrate the problem as:
3 one fourth units of 1 half unit and I could see an answer of 3 one eighth units so clearly.
NICE!

Over the past couple of years it seems I’ve been using unitizing language when talking about like terms. Rather than saying something like “4x plus 2x” I’ve been making a conscious effort to instead same something like “4 x’s plus 2 x’s”. It’s made a huge difference when I look at how few students will now write 4x+2x=6x^2. I always tell them just like you can’t add apples and oranges together, you can’t add apples to apples and get oranges.

Anytime we can create a visual to explain what is happening, especially with a fraction problem, it helps build conceptual understanding. I worked with a group of Grade 5 students last year and they really struggled with understanding the math they were doing with multiplying and dividing fractions. The idea of drawing an example of what they were doing was so new to them that it scared them. They were only taught the shortcuts of the algorithm tricks but obviously did not fully understand what they were doing. It was tough to continue working with them too because we were doing distance learning too. Missed opportunity.😦

This was a great lesson! I always struggled with showing anything more than just adding and subtracting fractions with manipulatives, now I can confidently show them decimals as part of a whole, and multiplying and dividing fractions.

Seeing the fractions broken down like that resonated with me…although I am still a little confused and will need some time to understand this fully.
One question I have…how does a teacher unpack a problem to break it down like this for students?

You may find that as this course goes on you could benefit from pumping the breaks and rewatching and retrying some of the prompts yourself. One of the biggest misses I made earlier in my career was not taking time to do the math myself in multiple ways / with multiple strategies. Only by doing the math will we be able to better understand and thus be able to provide a better learning experience for students.
What you see in many videos is what we might be looking for in student work as they complete a problem and then consolidate using student generated approaches. Obviously pacing what you consolidate now vs what you might want to come back to is a big planning decision.
Once you pick what you’re hoping students take away, how might we create a provocation for them to explore?


The fraction multiplication and division near the end of the video was great. I’ll be looking at it again when I start planning my fraction unit for my grade 8 students. We often rush to the algorithm but I also have a handful of students who want to know the why and the textbook explanation is not the clearest.

The fraction example was fantastic. Even as an 8th grade teacher this lesson is important to show to my students. I feel more smaller lessons of understanding and review such concepts that where supposedly taught in prior years are necessary to bring back. I state this because I still see students struggling with these standards and not fully understanding the reasoning behind why such concepts function the way they do.

I have middle schoolers struggling with multiplying and dividing with fractions. I am looking forward to using the visuals you showed to go with why our ‘methods/steps’ work. I just need to practice my verbage on it a bit more 🙂

The visuals were so powerful, especially with the multiplication and division of fractions. I appreciated Kyle’s advice to hold off on the algorithms and the symbolic use of numerators and denominators. It is so important for students to see the unit fractions inside of other fractions. This idea is something they have explored as young learners when they thought about what is inside the whole number 5 or 10. Composing and decomposing — it comes up again! But not when we rush to teach by algorithms and rules (that expire).
Quotitive vs. partitive — very important as well. I still have to cheat and look at my notes when I am teaching this!!! Drawing the models are so important because the representations you end us creating look different based on whether quotitive or partitive.

So glad you’re a supporter of slowing down and not rushing algorithms!
Also – that partitive and quotative fun can be so confusing at first. I always say to educators who think it is “too complex” or it will “confuse kids”… imagine how confusing it would be if we went through life not helping students realize that there IS two types of division? Imagine how much of a confusing struggle that would be?
So glad you’re diving into the learning with open arms!


I liked all the examples staring with basic unit and addressing almost many skills through visuals. This can build students’ conceptual understanding. I was simply enthused by the distributive property with units. I do have a question about how to use these resources directly with my students. Are these skill based video clips available for us as such so that we can show these to our students?
 This reply was modified 1 year, 10 months ago by Rajwant Cheema.

Not going to sugarcoat it here…I am still needing additional information on the two types of division represented in this lesson. I have come from a “How many of these can you get from those” standpoint or how many of these can you share with others if you have those. Hoping there is more so I can gain additional insight down the road here!

It sounds like you have always been aware of both types as the language you used honours both contexts/approaches to dividing. I promise the two types are going to keep coming back into play throughout the remainder of the course! 🙂


What resonates with me is the statement about becoming numerically flexible. My students don’t have that flexibility. They need to play more with numbers. They do not have that sense of decimals having a relationship with fractions. They prefer decimals because they don’t understand fractions when in fact fractions can be “friendlier” to use when operating in an equation.

As a special ed teacher I often have to find new ways to show concepts that the students don’t understand. I recently realized that some of my middle schoolers did not really understand what basic fractions mean. I liked relating fraction units to counting block units. This may help them understand what two one sixths means and that it is not larger than 1/2 or 3 one sixths.

So true! And the importance of the unit fraction can certainly help them understand what they are truly counting. Nice work!


I have often thought that fractions and algebraic terms follow a lot of the same rules when it comes to how to combine them using addition, but this lesson helped me to understand that both of these are really just following the same rules of addition in general! The fact that we leave off units in the earlier grades to make it “simpler” prevents students from making connections that could help them much more in the long run!

Great take aways here! More and more we find that trying to make math “simple” can be problematic in the long run. Math is complex and we must help students engage in the productive struggle to allow for those larger connections to be made.


I like how things were brought right back to its most simplified version. Rather than introduce imperial or metric systems, if students can understand the simplicity of units, the understanding of parts of something is more concrete. I think as teachers, we jump too quickly into assuming students understand what we are teaching.

Yes! Awesome stuff. Thanks for sharing. Glad that it resonated.


I appreciated the transition proposed from fractions to decimal numbers. I’m excited to try unitizing fractions more intentionally with my students. I find this one of the more difficult things to help students understand in grade 5, like many people who have recently replied to this post. I’m also excited to help students understand partitive thinking a little better with relation not only to fractions but division, as I find that even though they’ve been “doing” division for a couple of years by the time they come to grade 5, they struggle with applying their understanding to dividing 3digit numbers by 1digit numbers. I’d like to be able to teach this better so it is less about remembering the steps and more about understanding what they are doing.

The importance of highlighting units—especially with fractions is vital. In my experience students really can’t visualize what fractions are or what is happening when we operate upon them. Division of course is the most difficult to visualize.
Thanks for the examples!

So true! We have some fantastic fraction operations units that you should definitely check out. Here’s a great one:


The focus on units of measure is super helpful in thinking about Algebra and combining like terms. Students often struggle with the concept of what goes together when using variables. The idea of stressing that these are actually different units of measure will be helpful when teaching combining like terms.

The visual of partitive division is incredibly helpful. I’d love to see something like that for fraction division that is not with a unit fraction, but something like 3/4 divided by 4/5…

I love the visuals! My math 8 class is working on solving equations, so the algebraic example resonated with me, but I also liked the fraction and decimal examples because my students fall apart when they aren’t working with whole numbers. Tomorrow I am going to make sure to use clearer language as we work with decimals. I think that will help.

Fantastic to hear! We have a lot of problem based units relating to fractions and decimals here: https://learn.makemathmoments.com/tasks make sure to check them out!


The part that resonated with me was using the exact, correct notation/vocabulary.
Ex. One whole unit of 3 onefourths…then showing what it looks like symbolically.
I really also think we need to give ourselves permission as teachers to SLOW DOWN and make sure this is understood and solidified as we go through the teaching and learning of fractions.

Incorporating the visual when working with fractions is still needed. Even at grade 7. Teachers are rushed to ‘cover content’ that we forget that this concept needs time and practice. Fractions are hard to visualize, so explicitly showing them how to do so is another tool they can use to gain understanding and confidence.

Visual models are so key! We often assume because students are older that they can think abstractly. The reality is that the visual helps them to build their ability to think abstractly!


The examples of the unitizing of fractions with the concrete representation was extremely important to me. Before I taught sixth grade I taught fourth grade. I remember trying to make fractions less confusing when talking about like denominators. I would write fourths in words instead of the numeric representation. I started doing this after I had a math professional development. I didn’t understand at the time how we were making the fraction vocabulary more concrete and less symbolic. This is huge, and yet so simple. It really would make a difference to students if we would pull back on the symbolic and let them play with the models and fractions.
I am not sure where another example would be as I am still thinking through how I can use this idea with the intervention of my struggling students who are expected to come to me with symbolic understanding of the fractions but the reality is that they don’t.

Using words for fractions instead of the fraction bar initially is extremely helpful. Once students build an understanding of what fractions look and sound like, then the symbolic representation can be introduced as well. It is a complex concept and requires much time and effort to build that fluency!


We were using hanger diagrams to model balancing equations. The language of unit groups will be so helpful to this. Sometimes the diagram gives a value to a shape as 1 unit–say a hanging triangle is worth 1. Then it will change and a hanging triangle might be worth 3 so i triangle is 1 three unit group. Wow! The words here will be so powerful to student understanding. Thanks!

The example of why the unit of measure matters that I resonated with the most was the multiplication model where you can combine the same number of units to create groups of units.
In proportional table we are looking at unitizing the unit rate, which can also be created into groups of units.
Partition division is still hard to wrap my head around.

Don’t you worry – the two types of division takes a long time to feel comfortable with. Through this course, hopefully you’ll begin to notice and name them with confidence!
